Substitute the free field solutions (3-36) to the Klein-Gordon equation into the probability density operator relation (3-89) and then insert that into (3-90) to find the effective probability density operator expressed in terms of number operators (3-91). It will help you in doing so to note that for any term where k=k′, the destruction and creation operators will cause the ket to be different from (orthogonal to) the bra, so the resulting term in the expectation value ρˉ will be zero. Hence, those terms can be ignored in determining an effective ρ.
Note that the result you get is restricted to situations where all particles (in the ket) are in k eigenstates, which is almost invariably the case in QFT problems and applications. With particles in general (non k eigen) states, ρ becomes more complicated.